A better approximation of moments of the eigenvalues and eigenvectors of the sample covariance matrix

Lawley (Lawley, 1956) obtained an approximation, through the first terms of a series expansion, of certain moments of an eigenvalue of the sample covariance matrix. The aim of this paper is to improve that approximation and to calculate a similar approximation for certain moments of the associated eigenvector. The results have practical applications in certain fields of Statistics, such as Influence Analysis.